10 TRYING TO BARK, PURR WAY TO WONDERFUL TITLE

Ten of the nation`s most talented, terrific dogs and cats will compete for the coveted title of America`s Most Wonderful Pet on April 29 at Medinah Temple.

Among this year`s entries in this all-amateur talent extravaganza that is open to the public are:

Barney, a long-haired white cat who rides on his own cocktail tray while owner Kim Sloan of Houston shakes, rattles and rollerskates to the tune of

''Stray Cat Strut.'' Barney has been going for a spin since he was 6 months old; he`s now 8.

A Chicago ''Bear,'' a 4-year-old Keeshond owned by Sue Reiter, who loves to score big with the senior citizens he visits in nursing homes.

''Peanut,'' a 2 1/2-pound punked-out pup from Chula Vista, Calif., who loves to drive his own mini-hotrod wearing a Sherlock Holmes disguise. He has a little help at the wheel from owner Betty Lou Conley. Peanut is a Mypet, a new dog breed that combines the characteristics of the Maltese, Yorkshire terrier, poodle and Shih Tzu.

Rusty, a tabby cat from New York, who totally turns around the belief that ''you can`t train cats to do tricks.'' Rusty can sit, stay, lie down, roll over, shake hands and wave. He can even play the piano and patty-cake. In all, Rusty has mastered 36 tricks, says owner Robyn Volin.

Brute, a 3 1/2-pound Chihuahua from Vidor, Tex., has a bluegrass singing routine that makes him the Kenny Rogers of the canine set. Owner Claude Leflette says Brute, who appears in a crocheted cowboy hat, ''likes to perform at night so he can see the flashbulbs.''

Brandy and Willie, a terrier/golden labrador retriever and a collie mix, are canine exercise enthusiasts from Ontario, N.Y. Their coach is owner Bob Remis, and their mastery of dog aerobics includes a workout over hurdles and hoops. The dynamic duo also performs a hot new dance called ''Puppy Paws,''

which just may replace the ''Funky Chicken'' in popularity.

They and the other canine and feline finalists in the 1986 contest are winners of local competitions staged in animal shelters and humane societies from coast to coast.

All the dogs and cats were trained by their owners.

A panel of judges including celebrities, media representatives and animal experts will select the Most Wonderful Pet, as well as two runners-up and a Most Congenial.

Actor Kevin Dobson, of the TV series ''Knots Landing,'' will be emcee. Dobson is the national honorary chairperson for the PAW Council, contest sponsor. (Dobson also is known for his portrayal of Detective Crocker on CBS- TV`s ''Kojak.'' He and his wife, Susan, and their three children share a home in California with Crocker, an 11-year-old boxer.)

The winner of the Most Wonderful Pet contest and the shelter it represents will receive $1,000 each. Runners-up and their shelters will be awarded $500 each, and the recipient of the Most Congenial prize will receive $250.

The contest wraps up the nationally recognized Pets Are Wonderful Month, a campaign held each April to call attention to the thousands of deserving but homeless dogs and cats waiting in shelters to be adopted.

The Most Wonderful Pet contest will begin at 10:30 a.m. and end by noon at Medinah Temple, 600 N. Wabash Ave.

Free tickets can be picked up in advance from the Chicago Anti-Cruelty Society, at 157 W. Grand Ave., or will be sent by mail if ordered by phone, 644-8370.